Are there any hard and fast rules as to how to assign relative importance of one authority over another in formulating a psak halacha? For example if the Rambam and the Rashba argue over a specific halacha to whom would a posek give more credence and on what basis? Are there blanket generalities or would we, for example favor the Rambam in hilchos Shabbos and the Rashba in shechita?

More or less, besides the Yemenite community that has held like the RamBam for around 1k years, everyone goes by the Shulchan Aruch(with Rema for Ashkenazim). Beyond that you're really in the domain of the various meforshim on the Shulchan Aruch, i.e. machlokes rishonim tends not to be the main issues anymore, its typically shitas that are after Beis Yosef. As far as how the Beis Yosef compiled his sefer and the Rema and the various commentators, I think a lot had to do with their geographically linked customs, i.e. the Rosh was Ashkenazi as was his son, where as the Rambam was mizrachi etc.
– warz3Apr 2 '15 at 16:48

1 Answer
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Generally, (especially in arguments that take place in the Talmud), we generally view the older source as priority. In your case, Rambam would have spoken before Rashba. (Since Rambam died in 1204, and Rashba wasn't born until 1235, I make the assumption that Rashba argued with him posthumously, which happens often). So I would say, at the most general level, Rambam's word would be prioritized, but there are other factors that you should take into account that are specific to the text your referring to. But as a basic rule, who said it first?

However, as has been pointed out to me, הלכה כבתראי sates that within the same halachic period the newer source would be better, since whoever wrote the newer source has read more opinions than the person who wrote the older source.

@IsaacKotlicky What you said is probably true. Within the specific era the newer source could be considered more accurate, as הלכה כבתראי would state, since the newer source comes from someone who has heard more opinions. But between multiple periods, the older source is usually prioritized.
– ak11234Feb 25 '16 at 3:26